Russians were furious that Kharkiv was handed over without a fight. They feel humiliated even if militarily we all know that they didn't have enough men to fight there. Alternative media is saying that the Russian population wants Putin to step up and end this, but he also is not in a hurry because of winter.
I can't believe that I am about to do this, but I'm going to almost take Putin's side here!
I half-agree with your basic argument: Putin went into this invasion with too few troops, because he expected that his Army would be viewed as liberating heroes and that the Ukrainian government would collapse inside 72 hours.
There is a comparison with Argentina invading the Falkland Islands in 1982 because they had misinterpreted a number of British actions such as the retirement of HMS Endurance as evidence that Britain would not fight back but would negotiate.
To Putin's surprise the Ukrainian people - even the Ukrainian Russians - did not welcome the Russian Army, and full-scale resistance occurred. But the Russian Army did not have the logistics to sustain the drive towards Kyiv, and their tanks and trucks with dodgy tyres and insufficient fuel broke down and he effectively lost a huge part of his best troops and material, before making fake claims that it was just a feint all along.
For the next six months he has lacked the advanced weapons to obtain a clean American-style victory and has instead been reduced to using WW2 artillery tactics to flatten Ukrainian towns and cities and to grind out tiny areas of advancement, but at the cost of completely destroying the lives - and support - of the very people he pretends to be liberating.
Putin has probably lost half his professional army, 90% of their advanced weaponry and the support of Ukrainian Russians, and he now has an overstretched army with weak points all over the south, east and northeast of Ukraine.
That is not to say that the Ukrainians have not suffered heavy losses too. But they are better trained and they increasingly are replacing their spent weaponry with superior western weapons, while Russia is reduced to shopping in Tajikstan, North Korea and Iran to try to replace its spent weaponry.
Russia's best troops and weapons are now defending Kherson, which is the only major place that they have conquered all year. They are making tiny progress at a single Donbas location - Bakhmut - while steadily losing what they had previously seized everywhere else in Donbas.
Putin is stuck. He hasn't got enough troops or modern weapons to hold all three of Donbas, Crimea and Kherson and that is after his Army had to retreat suddenly from Kharkiv, running away so fast that they left behind vast amounts of material.
The thing is, Biden is not going for broke by providing Ukraine with the weapons that it needs to finish the war off quickly. Part of this is political - there has been virtually no opposition to gifting Ukraine the most obsolete 1% of the USA's weapons, but giving them more advanced weaponry may not enjoy bipartisan support. But more importantly, every day that the war continues Russia's military is damaged further and it loses more influence in the former Soviet Republics. Already Kazakstan is democratising, and the protagonists in the Tajik-Kyrgyz and Armenaia/Azeri conflicts are brazenly ignoring Russia's attempts to intervene.
I don't think most Russians understand that they have just lost at Kharkiv. The elites know, and may be angry, but the diet of stultifying propaganda fed to the general public means that they think that their brave boys are still rooting out evil Nazis across Ukraine, and being welcomed as liberators as they do it.
The Army and political elite know that they can't win this war now, and that there is a genuine danger of losing what they already held in Crimea and Donbas. But I don't think that the Russian public knows that.
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